Psychometric Functions & the PSE
Psychophysics is the science of turning perception into numbers: how bright a light must be before you notice it, how much one tone must rise before it sounds louder, how long an interval must last before it feels long. The instrument behind every such measurement is the psychometric function, an S-shaped curve that links a stimulus you control to the responses an observer gives. Fitting that curve, and reading two numbers off it, is the whole analysis, and the machinery is one you already met in logistic regression.
Psychophysics and the method of constant stimuli
Psychophysics measures the link between a physical stimulus and the perception it produces, a program Gustav Fechner launched in 1860 and one often credited as the start of experimental psychology. Its most transparent procedure is the method of constant stimuli. You fix a ladder of stimulus levels spanning the interesting range, from a level almost everyone judges one way to a level almost everyone judges the other, present each level many times in random order, and record the proportion of one response at each. Plotted against stimulus level, those proportions trace the psychometric function. (Adaptive procedures such as staircases reach the threshold with fewer trials by tuning each stimulus to the last response, but constant stimuli is the one to learn first, because it lays the entire curve bare.)
That single recipe answers a remarkable range of questions. In vision: the faintest contrast at which a pattern becomes visible, or which of two lines looks longer. In hearing: the quietest detectable tone, the measurement behind the audiogram in every hearing clinic, or which of two tones is louder. In touch: the smallest gap at which two pressed points stop feeling like one. In the lifted-weight experiments that gave us Weber's law: which of two weights feels heavier. In the perception of time: whether an interval feels short or long. Detection tasks ask whether a stimulus is present at all; discrimination tasks ask which of two is more intense. Both yield the same shape, a proportion climbing smoothly from near 0 to near 1 as the stimulus grows.
The psychometric function
Whatever sense is being probed, that climbing proportion is the psychometric function: stimulus level in, response probability out. The analysis job is to describe its smooth rise with a compact curve and to summarize that curve with a few parameters.
Three families do most of the work in practice. The logistic is the same S-curve that powers logistic regression. The cumulative Gaussian comes from classic signal detection theory: it is what you get if the observer's internal reading of the stimulus is the true level plus normal noise. The Weibull is asymmetric on the stimulus axis and is a staple of detection tasks. Each has two parameters, a location and a spread, and each is fit by maximum likelihood on the raw trial counts. A trial is a Bernoulli outcome whose probability depends on the stimulus, which is exactly the binomial machinery of a GLM.
To make this concrete, the rest of the lesson stays with one task: the temporal bisection procedure, a standard way to study how people perceive duration. The observer first learns two anchor durations, a short one (200 ms) and a long one (800 ms). Probe intervals of in-between lengths then arrive one at a time, and each judgment is simply whether the probe felt closer to short or closer to long. The share of “long” responses at each probe duration is precisely the proportion the method of constant stimuli collects, and the interactive below fits a curve to it.
🎮 The Bisection Experiment
Each dot is the share of “long” responses at one probe duration; the curves are maximum-likelihood fits, with dashed drops at each condition's PSE. The simulated observer is always Gaussian; the selector changes only the fitted model. Sliders re-analyze the same frozen trials; “New trials” draws fresh ones.
Two numbers: PSE and JND
The point of subjective equality (PSE) is the stimulus level at which the two responses are equally likely, the 50% point of the fitted curve. In bisection it answers a genuinely psychological question: which duration feels like the exact midpoint between the anchors? Not necessarily the arithmetic middle. Timing studies often find bisection points near the geometric mean of the anchors (for 200 and 800 ms that is √(200 × 800) = 400 ms, not 500), one of the classic clues that our sense of time is compressive.
The just-noticeable difference (JND) measures precision: half the distance between the curve's 25% and 75% points. A steep curve means a small JND and a sharp sense of duration; a shallow curve means a large one. The two numbers are separate dials. One observer can be biased but precise (shifted PSE, steep slope), another unbiased but noisy. Dividing the JND by the PSE gives the Weber fraction, which lets you compare precision across different stimulus ranges.
Shift or slope: two different findings
Now manipulate something. Play the probe tones louder, make a visual stimulus brighter, add a distracting task. When a condition changes the curve, it can do so in two cleanly different ways. A PSE shift slides the whole curve sideways: more intense stimuli tend to be judged longer, so a brighter probe moves the curve left and the PSE drops while the slope stays put. A slope change flattens or sharpens the curve without moving its midpoint: the observer's timing has become noisier or more precise, and the JND grows or shrinks accordingly.
That dissociation is usually the finding itself. “The manipulation changed how long things seem” is a PSE story; “the manipulation changed how well people can time at all” is a JND story. The comparison condition in the interactive lets you dial each one independently and watch what the fits report.
Does the choice of function matter? Less than students fear, though not equally for every family. The two symmetric curves, logistic and cumulative Gaussian, pin their 50% point to where the data bite hardest, so their PSEs agree to a millisecond or two. The asymmetric Weibull can sit several milliseconds off, because it is really a detection function anchored at zero stimulus and its lopsided shape fits symmetric bisection data a little awkwardly. The spread parameter is shakier still: a logistic fit to Gaussian data reports a JND about 10% smaller, and the families diverge most in the tails, where lapses live. Pick the family your subfield uses, report it, and keep it constant across conditions.
Fitting it in practice
For a clean two-parameter fit you can stay inside logistic regression: model each trial's response from the stimulus level, and the coefficients hand you both numbers: PSE = −b₀/b₁ and JND = ln 3 / b₁. Comparing conditions is one model away: add the condition and its interaction with stimulus level, and the condition term tests a PSE shift while the interaction tests a slope change. Real laboratory data usually deserve one more ingredient: lapse rates, small probabilities of stray finger errors that keep the curve from quite reaching 0 and 1. Purpose-built tools such as psignifit and quickpsy estimate those alongside the curve; the “Try it yourself” box below shows both routes.
In a full study you fit the function per participant and condition, then treat each participant's PSE or JND as a single data point in a familiar analysis: a paired t-test across two conditions, or a mixed model when the design grows.
Why it matters: psychometric functions are the workhorse of psychophysics, and they appear anywhere a binary judgment meets a graded stimulus: perception labs, audiology clinics fitting hearing thresholds, vision screening, animal timing studies. If you can fit one S-curve and read off its PSE and JND, you can quantify what an observer perceives, and argue about bias versus sensitivity with numbers instead of adjectives.
Problem 34 of the practice problems reads a psychometric function off its logistic coefficients, the point of subjective equality and the just-noticeable difference, then compares how a quieter probe shifts both.
Common questions
What is the difference between the PSE and the JND?
The PSE locates the curve; the JND measures how steeply it rises. A shifted PSE means the observer's subjective midpoint moved (a bias in perception), while a larger JND means discrimination got noisier (a loss of precision). They come from the same fit but answer different questions, and a manipulation can change either one without touching the other — that dissociation is often the headline result of a psychophysics study.
Which function should I fit: logistic, cumulative Gaussian, or Weibull?
The two symmetric families (logistic and cumulative Gaussian) put the PSE in essentially the same place, and the asymmetric Weibull is usually within a handful of milliseconds; the spread parameter (and with it the JND) varies more, and the tails differ most. The cumulative Gaussian has the cleanest signal-detection interpretation (internal noise is normal), the logistic is computationally convenient and matches logistic regression, and the Weibull suits detection tasks where performance is anchored at zero stimulus. Convention in your subfield is a fine tie-breaker; just report the family and keep it constant across conditions.
Do I need a lapse rate in my psychometric model?
For real observers, usually yes. People blink, press the wrong key, and drift off, so a few errors appear even at the easiest stimulus levels. A two-parameter fit has to tilt the whole curve to accommodate those trials, which biases the slope (and so the JND); adding a small lapse parameter, either fixed at something like 0.02 or estimated with an upper bound, absorbs them instead. Tools built for psychophysics (psignifit, quickpsy) include lapse and guess rates by default, which is a good reason to graduate to them once the basic fit makes sense.